**30**

votes

**4**answers

2k views

### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...

**27**

votes

**2**answers

2k views

### Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**9**

votes

**7**answers

4k views

### Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...

**10**

votes

**2**answers

704 views

### Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?

**58**

votes

**3**answers

4k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...

**34**

votes

**2**answers

2k views

### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...

**9**

votes

**1**answer

334 views

### The Simultaneous Conjugacy Problem in the symmetric group $S_N$

We are interested in the following notions in the case $G=S_N$, the symmetric group on
$\{1,\dots,N\}$.
Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define
...

**6**

votes

**2**answers

1k views

### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...

**5**

votes

**1**answer

524 views

### What is the largest number of k-element subsets of a given n-element set S such that…

Given a set S of n elements. What is the largest number of k-element subsets of S such that every pair of these subsets has at most one common element?

**5**

votes

**1**answer

463 views

### Difference Sets

Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...

**59**

votes

**17**answers

45k views

### Google question: In a country in which people only want boys [closed]

Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
...

**55**

votes

**9**answers

6k views

### Why do Bernoulli numbers arise everywhere?

I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...

**57**

votes

**5**answers

2k views

### Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer.
A polyomino is usually defined to ...

**41**

votes

**2**answers

5k views

### Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly Orthogonal to Morse
!
Harold Calvin Marston Morse (24 March ...

**31**

votes

**5**answers

3k views

### Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...

**30**

votes

**20**answers

3k views

### Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...

**35**

votes

**6**answers

2k views

### Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime

This is a question first I asked in SE but since there was no suggestion or solution, I decide to put it here.
Consider an $n\times n \times n$ Cube containing $n^3$ unit cubes. Is it possible to ...

**25**

votes

**2**answers

1k views

### Is there a finite family of functions such that the max of any two functions can be dominated by a third?

Is it true that for every $t$ there is an $n$ and there exists a finite function
family, $\cal F$, whose members are from $[n] \to \mathbb N$ (taking all different
values) and for any $f_1, \ldots, ...

**15**

votes

**10**answers

3k views

### Applications of infinite Ramsey's Theorem (on N)?

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...

**20**

votes

**1**answer

1k views

### Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order ...

**14**

votes

**5**answers

1k views

### A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...

**19**

votes

**2**answers

856 views

### Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book ...

**40**

votes

**4**answers

3k views

### Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...

**10**

votes

**5**answers

3k views

### infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...

**21**

votes

**3**answers

2k views

### How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot ...

**6**

votes

**1**answer

1k views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

**15**

votes

**2**answers

790 views

### Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...

**10**

votes

**3**answers

936 views

### A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le ...

**9**

votes

**2**answers

733 views

### Is the Steiner ratio Gilbert–Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest
network interconnecting $P$ must be a tree, which is called a Steiner minimum ...

**16**

votes

**3**answers

1k views

### Does subgroup structure of a finite group characterize isomorphism type?

Question
Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...

**7**

votes

**2**answers

747 views

### Highbrow interpretations of Stirling number reciprocity

The number ${n \choose k}$ of $k$-element subsets of an $n$-element set and the number $\left( {n \choose k} \right)$ of $k$-element multisets of an $n$-element set satisfy the reciprocity formula
...

**6**

votes

**4**answers

2k views

### Mean minimum distance for N random points on a unit square (plane)

A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...

**5**

votes

**2**answers

221 views

### Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A ...

**3**

votes

**1**answer

1k views

### Counting graphs

I'd love your help with this question.
Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $p$ vertices and $q$ edges are there where $p+q=n$?
Thank you very much.

**7**

votes

**1**answer

361 views

### maximum size of intersecting set families

Suppose $n$ is a big number and $k\geq 2$. How many sets $S_1,\dots,S_m\subset [n]$ can we find such that
(1) $|S_i| = k$ for all $i$,
(2) $|S_i\cap S_j| \leq 1$ for all $i\ne j$.
What's the maximum ...

**7**

votes

**1**answer

195 views

### The time to drift a binary string from one state to another via deterministic selection of two possible random bit mutation procedures

I have some length $L$ binary string consisting of an ordered array of bits: $(b_1, b_2, ..., b_{L})$, where all bit values $b_i$ are originally set to zero. I'd like a particular set of $n$ bits to ...

**6**

votes

**1**answer

253 views

### Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...

**5**

votes

**2**answers

343 views

### Generalisations of Petersen's 2-factor theorem?

Petersen's 2-Factor Theorem (1891): A $(2r)$-regular graph can be decomposed into $r$ edge-disjoint $2$-factors.
I'd like to use this theorem (or a more general version of this theorem) to imply ...

**4**

votes

**1**answer

184 views

### Graphs with constant edge imbalance

The imbalance of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997)
I'm interested in the set of ...

**2**

votes

**3**answers

1k views

### Estimating a partial sum of weighted binomial coefficients

There is a well-known estimate for the sum of all binomial coefficients $\binom{n}{k}$ satisfying $k \leq \alpha n$ for some $\alpha$ satisfying $0 < \alpha \leq 1/2$:
$$ \sum_{k=0}^{\alpha ...

**1**

vote

**1**answer

185 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**9**

votes

**5**answers

1k views

### A sum involving derivatives of Vandermonde

Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$.
I am intersted in the calculation of the following expression for fixed $k$:
$$\sum_i (x_i)^k (d/dx_i)^k V(x_1 , ...

**8**

votes

**2**answers

745 views

### Sperner's lemma and Tucker's lemma

In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proof that Tucker's ...

**3**

votes

**1**answer

149 views

### Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...

**3**

votes

**1**answer

551 views

### Estimate on sum of squares of multinomial coefficients

I am interested in approximating the sum of the squares of the multinomial coefficients, i.e.
$a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$
or more general,
...

**2**

votes

**1**answer

293 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...

**2**

votes

**0**answers

651 views

### Necklaces and the generating function for inversions

The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated:
http://mathworld.wolfram.com/Necklace.html
Let us ...

**1**

vote

**0**answers

128 views

### Non-trivial lower bound on the number of “Graph Diagonals”

The definition of Graph Diagonals, that are the subject of this question, is based on the notions of crossing edges and on connected graphs:
Two edges $AC$ and $BD$ of a complete, symmetric and ...

**78**

votes

**4**answers

3k views

### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
...

**25**

votes

**15**answers

7k views

### Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...